docs: editor comments on the paper (#44)

Author: olexandr-konovalov

paper.md

@@ -27,14 +27,14 @@ This is a highly specialised software aimed mostly for computational mathematici
 
 # Key features
 
-- As a header-only *C++* template code it's greatest advantage is the combination of speed, generic programming and convenience for the end user. Open Source license together with template specialisation mechanism allows contributors to add-in support for custom objects, define specific functions and extend the scope of the library.
+- As a header-only *C++* template code, it's greatest advantage is the combination of speed, generic programming, and convenience for the end user. Open Source license together with template specialisation mechanism allows contributors to add support for custom objects, define specific functions and extend the scope of the library.
 - The most important specialisation, already included in the library itself, is the introduction of operations in hypercomplex algebras over truncated polynomial rings. These allow for many cryptographic applications as described in a dedicated section below. 
 - Another template class specialisation introduces the support for arbitrary high precision of calculations via GNU MPFR library [@fousse:inria-00070266], for which the operators have been overloaded such that all the instructions are carried out on specific data structures.
 - State of the art technology for software engineering:
   - CI/CD mechanism set up with GitHub Actions: automatic tests for library installation, source code inclusion, compilation and execution,
   - extensive unit testing with Catch2 framework [@catch2] alongside code coverage measurement uploaded to Codecov; current coverage: 100%,
   - source code linting with cpplint [@cpplint] - Google code style enforced,
-  - automatic documentation generation and hosting on GitHub Pages: build via Doxygen [@doxygen], publishing via Actions.
+  - automatic documentation generation and hosting on GitHub Pages: build via Doxygen [@doxygen], publishing via GitHub Actions.
 
 # Cryptographic applications
 
@@ -47,16 +47,16 @@ Every element of $\mathcal{R}$, $\mathcal{R}_p$, $\mathcal{R}_q$ may be writted
 f = \sum_{i=0}^{N-1} f_i x_i \equiv [f_0, \ldots ,f_{N-1}]
 \end{equation}
 
-Addition operation $+$ refers to a regular element-wise addition of coefficients (modular for $\mathcal{R}_p$ and $\mathcal{R}_q$).
+where the addition operation $+$ refers to a regular element-wise addition of coefficients (modular for $\mathcal{R}_p$ and $\mathcal{R}_q$).
 Multiplication $\star$ within this structure is defined as:
 
 \begin{equation}\label{eq:ringmul}
 f \star g = \sum_{i=0}^k f_i g_{k-i} + \sum_{i=k+1}^{N-1} f_i g_{N+k-i}
 \end{equation}
 
-With a final reduction modulo $p$ or $q$ in the modular quotient rings.
+with a final reduction modulo $p$ or $q$ in the modular quotient rings.
 
-Based on the above let us pick an integer $\lambda \geq 0$ and define three corresponding algebras, generated by the Cayley-Dickson process:
+Based on the above, let us pick an integer $\lambda \geq 0$ and define three corresponding algebras, generated by the Cayley-Dickson process:
 
 \begin{equation}\label{eq:algebras}
 \begin{aligned}
@@ -66,10 +66,11 @@ Based on the above let us pick an integer $\lambda \geq 0$ and define three corr
 \end{aligned}
 \end{equation}
 
-Note that $\forall x\in \mathcal{A^\lambda}: x = (a, b) | a, b \in \mathcal{A^{\lambda-1}}$.
+where the ddition operation $+$ refers to ring addition defined above.  
 
-Addition operation $+$ refers to ring addition defined above.  
-Multiplication $\times$ is defined recursively based on conjugation operation $^*$ as below:
+Note that $\forall x\in \mathcal{A^\lambda}: x = (a, b)$, where $a, b \in \mathcal{A^{\lambda-1}}$.
+
+Multiplication $\times$ is defined recursively based on the conjugation operation $^*$ as below:
 
 \begin{equation}\label{eq:recursivemultiplication}
 \begin{aligned}
@@ -83,10 +84,10 @@ Multiplication $\times$ is defined recursively based on conjugation operation $^
 \end{aligned}
 \end{equation}
 
-Which holds for the modular algebras too given a final reduction modulus $p$ or $q$.
+which as well holds for the modular algebras, given a final reduction modulus $p$ or $q$.
 
-Based on the above let us define a general scheme for
-hypercomplex-based cyptosystems. Having agreed on $(N, p, q)$ Bob
+Based on the above, let us define a general scheme for
+hypercomplex-based cyptosystems. Having agreed on $(N, p, q)$, Bob
 selects $F, G \in \mathcal{A^\lambda} : \exists F_p^{-1}\in\mathcal{A_p^\lambda} \wedge \exists F_q^{-1}\in\mathcal{A_q^\lambda}$.  
 A procedure to generate the public key $H\in\mathcal{A_q^\lambda}$ is then given by:
 
@@ -112,12 +113,12 @@ The following decryption consist of three steps:
 \end{aligned}
 \end{equation}
 
-If the decryption was successfull Bob receives $D_3 = M$ (up to coefficients' centered lift in $\mathcal{A_p^\lambda}$).
-Please remember that lattice-based cryptography is always burdened with a chance of decryption failure due to incorrect recovery of polynomial's coefficients.
-Also, for $\lambda \geq 4$ note that $\mathcal{A^\lambda}$ is not alternative
-nor associative thus successful decryption relies on a careful initial choice of $F$
+If the decryption was successfull, Bob receives $D_3 = M$ (up to coefficients' centered lift in $\mathcal{A_p^\lambda}$).
+Please remember that the lattice-based cryptography is always burdened with a chance of decryption failure due to an incorrect recovery of polynomial's coefficients.
+Also, for $\lambda \geq 4$ note that $\mathcal{A^\lambda}$ is neither alternative
+nor associative; thus successful decryption relies on a careful initial choice of $F$
 (e.g. $F: \exists! i\in\{0, \ldots ,2^\lambda-1\}: F_i \neq 0$).
-For a more detailed deriviation of similar
+For a more detailed coverage of similar
 cryptosystems please see publications
 presenting QTRU[@QTRU] and OTRU[@OTRU].
 
@@ -141,21 +142,21 @@ All of the data and code required to reproduce these results is available in the
 
 # State of the field
 
-When it comes to a general hypercomplex framework the well-known _boost C++_ libraries deserve the most notable mention here [@boost]. Unfortunately their scope is limitted as they only provide quaterions and octonions classes (however as an upside - all the operations are well optimised). Moreover, these libraries do not support operations on MPFR types natively. It may also be worth to mention the existence of smaller repositories like: [@quaternions] or [@cd], but, unlike our work, they often lack proper test suites, code coverage reports, documentation and are also significantly restricted in functionality which is a major drawback.
+When it comes to a general hypercomplex framework the well-known _boost C++_ libraries deserve the most notable mention here [@boost]. Unfortunately their scope is limited as they only implement classes for quaterions and octonions (however, as an upside, all the operations are well optimised). Moreover, these libraries do not support operations on MPFR types natively. It may also be worth to mention the existence of smaller projects like [@quaternions] or [@cd], but, unlike our work, they often lack proper test suites, code coverage reports, documentation and are also significantly restricted in functionality which is a major drawback.
 
 However, (most importantly) to our best knowledge there is currently no high-quality open-source library which natively supports cryptosystems based on truncated polynomial rings.
-Previous research described distinct versions of NTRU [@NTRU], among others: 4-dimensional QTRU [@QTRU], 8-dimensional OTRU [@OTRU]; some proposed 16-dimenisional STRU [@STRU], which correctness has not yet been verified.
-Despite these efforts no generalization has been provided yet.
-Our work is a first to: present that these procedures are vaild in arbitrary-high-dimensional Cayley-Dickson algebras (provided a careful choice of parameters of the system)
+Previous research described distinct versions of NTRU [@NTRU], among others: 4-dimensional QTRU [@QTRU], 8-dimensional OTRU [@OTRU]; and a proposed 16-dimenisional STRU [@STRU], correctness of which has not yet been verified.
+Despite these efforts, no generalization has been provided yet.
+Our work is a first to present that these procedures are vaild in arbitrarily high-dimensional Cayley-Dickson algebras (provided a careful choice of parameters of the system)
 and to provide reproducible examples of a successful encryption/decryption procedures.
 Finally, it has not escaped our notice that the specific polynomial-based hypercomplex multiplication scheme we presented immediately suggests a possible hashing mechanism for string messages.
 
 # Acknowledgments
 
-We would like to express our wholehearted gratitidue towards: the members of
-a facebook group _>implying we can discuss mathematics_, who aided us
+We would like to express our wholehearted gratitude towards: the members of
+a Facebook group _>implying we can discuss mathematics_, who aided us
 with clarifications and suggestions related to the topic of research
-as well as a _Cryptography Stack Exchange_ user: _DanielS_, who helped us
+as well as a _Cryptography Stack Exchange_ user _DanielS_, who helped us
 analyse and understand specifics of lattice-based cryptosystems.
 
 # References